Yes, that's the point. The OP was claiming thathe had invented a form of generalized continued fractionthe produced best simultaneous ? rational approximationsand was periodic for higher irrationalities.It turns out that he was not claiming that he could produceall best rational approximation but simply thatthe occasional best approximation might occur among the"convergents".

He now says that his GCFs produce periodic expansions ofhigher irrationalities in some way based onBernoulli's dominant zero method.

But this is not what is usually meant byperiodicity. If every algebraic number has a "periodicexpansion" in the sense that it is some function of thecoefficients of its characteristic polynomial, thenthere is no way of discriminating between irrationalitiesof various degrees.

Essentially, all the OP is doing is expressingthe ratio of the nth and n-1th terms in a recurrence relationas a complicated fraction involving the coefficientsof the characteristic polynomial.I would not have thought that this approach would produce periodicity,except in this trivial sense, or any best rational approximations.

Finding solutions to the cubic Pell is easy (use Pari/gp).

Do you mean systematically using generalizations of continuedfractions orother methods for finding cubic units or just intelligenttrial and error ?

Proving (unconditionally) the solution you find is, say, thefundamentalsolution can take a bit more effort.

Yes, I saw your recent post and Israel's interesting answer.Have you found some way of estimating an upper boundfor the power of the unimodular matrix so that you canfind the fundamental unit after a small number of trials ?